0,8

Also number of standard Young tableaux with last row of length k.

Joerg Arndt and Alois P. Heinz, Rows n = 0..60, flattened

Wikipedia, Young tableau

Triangle starts:

00: 1;

01: 0, 1;

02: 0, 1, 1;

03, 0, 3, 0, 1;

04: 0, 7, 2, 0, 1;

05: 0, 20, 5, 0, 0, 1;

06: 0, 56, 14, 5, 0, 0, 1;

07: 0, 182, 35, 14, 0, 0, 0, 1;

08: 0, 589, 132, 28, 14, 0, 0, 0, 1;

09: 0, 2088, 399, 90, 42, 0, 0, 0, 0, 1;

10: 0, 7522, 1556, 285, 90, 42, 0, 0, 0, 0, 1;

11: 0, 28820, 5346, 1232, 165, 132, 0, 0, 0, 0, 0, 1;

12: 0, 113092, 21515, 4378, 737, 297, 132, 0, 0, 0, 0, 0, 1;

13: 0, 464477, 82940, 17082, 3003, 572, 429, 0, 0, 0, 0, 0, 0, 1;

...

The T(6,2)=14 ballot sequences of length 6 with 2 maximal elements are (dots for zeros):

01: [ . . . . 1 1 ]

02: [ . . . 1 . 1 ]

03: [ . . . 1 1 . ]

04: [ . . 1 . . 1 ]

05: [ . . 1 . 1 . ]

06: [ . . 1 1 . . ]

07: [ . . 1 1 2 2 ]

08: [ . . 1 2 1 2 ]

09: [ . 1 . . . 1 ]

10: [ . 1 . . 1 . ]

11: [ . 1 . 1 . . ]

12: [ . 1 . 1 2 2 ]

13: [ . 1 . 2 1 2 ]

14: [ . 1 2 . 1 2 ]

The T(8,4)=14 such ballot sequences of length 8 and 4 maximal elements are:

01: [ . . . . 1 1 1 1 ]

02: [ . . . 1 . 1 1 1 ]

03: [ . . . 1 1 . 1 1 ]

04: [ . . . 1 1 1 . 1 ]

05: [ . . 1 . . 1 1 1 ]

06: [ . . 1 . 1 . 1 1 ]

07: [ . . 1 . 1 1 . 1 ]

08: [ . . 1 1 . . 1 1 ]

09: [ . . 1 1 . 1 . 1 ]

10: [ . 1 . . . 1 1 1 ]

11: [ . 1 . . 1 . 1 1 ]

12: [ . 1 . . 1 1 . 1 ]

13: [ . 1 . 1 . . 1 1 ]

14: [ . 1 . 1 . 1 . 1 ]

These are the (reversed) Dyck words of semi-length 4.

b:= proc(n, l) option remember; `if`(n<1, x^l[-1],

b(n-1, [l[], 1]) +add(`if`(i=1 or l[i-1]>l[i],

b(n-1, subsop(i=l[i]+1, l)), 0), i=1..nops(l)))

end:

T:= n->`if`(n=0, 1, (p->seq(coeff(p, x, i), i=0..n))(b(n-1, [1]))):

seq(T(n), n=0..12);

# Alternative: counting SYT:

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+

add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)

end:

g:= proc(n, i, l) `if`(n=0 or i=1, h([l[], 1$n])*x^`if`(n>0, 1,

`if`(l=[], 0, l[-1])), g(n, i-1, l)+

`if`(i>n, 0, g(n-i, i, [l[], i])))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(g(n, n, [])):

seq(T(n), n=0..12);

b[n_, l_List] := b[n, l] = If[n<1, x^l[[-1]], b[n-1, Append[l, 1]] + Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; T[n_] := If[n == 0, 1, Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, {1}]]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 07 2015, translated from Maple *)

(PARI) (A238123(n, k)=if(k, vecsum(apply(p->n!/Hook(Vecrev(p)), select(p->p[1]==k, partitions(n, [k, n])))), !n)); Hook(P, h=vector(P[1]), L=P[#P])={prod(i=1, L, h[i]=L-i+1)*prod(i=1, #P-1, my(D=-L+L=P[#P-i]); prod(k=0, L-1, h[L-k]+=min(k, D)+1))} \\ M. F. Hasler, Jun 03 2018

The terms T(2*n,n) are the Catalan numbers (A000108).

Columns k=0-10 give: A000007, A238124, A244099, A244100, A244101, A244102, A244103, A244104, A244105, A244106, A244107.

Row sums give A000085.

Cf. A026794.

Sequence in context: A383097 A180049 A244454 * A128311 A334076 A132884

Adjacent sequences: A238120 A238121 A238122 * A238124 A238125 A238126

nonn,tabl

Joerg Arndt and Alois P. Heinz, Feb 21 2014

approved